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SPINAL -- Scaling-law and Preference Integration in Neural Alignment Layers

Arion Das, Partha Pratim Saha, Amit Dhanda, Vinija Jain, Aman Chadha, Amitava Das

arXiv:2601.06238v1cs.LGcs.AIcs.CL

TL;DR

The geometric footprint of preference-based alignment remains poorly understood, despite alignment being treated primarily as an output property. SPINAL traces layerwise structural changes and finds that DPO alignment concentrates in a terminal calibration zone within final decoder blocks.

  • Problem

    The geometric changes induced by preference optimization remain poorly understood beyond models’ outputs.

  • Method

    SPINAL diagnoses alignment as a depth-indexed trajectory using coupled spectral-scaling and semantic-transition signals.

  • Results

    DPO alignment concentrates in a terminal calibration zone within the final decoder blocks, with stronger signatures reflecting sharper representations, contracted transport, and coherent terminal trajectories.

  • Takeaways & Limitations

    SPINAL provides a mechanistic audit signal for alignment strength and localization, rather than a measure of downstream safety or utility.

  • Takeaways & Limitations

    Current evidence is limited to decoder-only transformers in a moderate 1.3B–13B parameter range, so broader architectural and scale generalization remains unestablished.

Abstract

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Direct Preference Optimization (DPO) is a principled, scalable alternative to RLHF for aligning large language models from pairwise preferences, but its internal geometric footprint remains undercharacterized, limiting audits, checkpoint comparisons, and failure prediction. We introduce SPINAL (Scaling-law and Preference Integration in Neural Alignment Layers), a diagnostic that measures how alignment reshapes representations across depth by tracing localized structural change layer by layer. Across model families, DPO produces a layerwise calibration effect concentrated in the final decoder blocks (often layers 21-30), where preference gradients most directly affect the next-token distribution. SPINAL encodes each checkpoint as a depth trace over (layer index, contraction score, transport score). The contraction score summarizes how quickly the tail of a layer's spectrum decays (how fast small modes vanish); higher values indicate stronger contraction into fewer effective directions. The transport score summarizes how much the token distribution shifts between adjacent layers using a bounded overlap measure; lower values indicate shorter, smoother steps through representation space. Aligned checkpoints show a late-layer ramp-up in contraction and a smooth reduction in transport, consistent with tightened and stabilized policy mass, while unaligned models trace higher-curvature, more entropic, and geometrically incoherent depth paths. Overall, alignment is geometrically localized: the final layers encode the dominant preference-induced corrections. SPINAL turns this localization into a practical audit signal, quantifying where alignment concentrates, how strongly it manifests, and when it begins to destabilize during training.

1 Alignment as Geometric Calibration: The SPINAL Hypothesis

The SPINAL hypothesis treats alignment as a geometric calibration of transformer depth, not merely an output property. DPO produces a localized shift in upper decoder layers, combining spectral sharpening with more structured transitions.

  • Central contribution: DPO induces a localized geometric shift in upper decoder blocks, where abstraction sharpens into decision.SPINAL traces this shift with the layerwise trajectory (ℓ, αℓ, Lℓ).
  • Spectral scaling: For ℓ > 20, aligned checkpoints show a monotonic rise in spectral tail exponent αℓ, revealing spectral sharpening weak or absent in base models.The spectrum follows ρ(σ) ∼ σ^-αℓ, with αℓ capturing compression and inductive bias.
  • Thermodynamic length: In aligned models, thermodynamic length Lℓ contracts in the upper block, indicating lower-entropy and more structured adjacent-layer transitions.Lℓ measures semantic effort between adjacent layers using Fisher geometry.
  • Geometric alignment zone: The geometric alignment zone captures net spectral sharpening plus semantic contraction in the last ∼10 layers.This localization is summarized through layerwise geometric fingerprints g_base(ℓ) and g_DPO(ℓ).
  • Geometric alignment zone: Across all studied LLMs, ∆align > 0, establishing alignment localization as a robust geometric signature of DPO-style alignment.Figures 1 and 8 visualize this transition.

2 What Is New in SPINAL? Relation to Prior Work

SPINAL advances prior localization observations by introducing a geometry-first, layer-resolved diagnostic that makes preference alignment quantitative, comparable, and auditable across model families. It captures a reproducible depth-localized alignment law through terminal-layer calibration and a compact statistic such as ∆align.

  • From Localization to Geometry: SPINAL converts observations that safety or alignment may be shallow or localized into a measurable geometric signature.Its geometry-first, layer-resolved design makes preference alignment quantitative, comparable, and auditable across model families.
  • Auditable Alignment Signature: SPINAL provides a reproducible, depth-localized law of preference alignment and a compact across-LLMs statistic such as ∆align.The statistic quantifies where calibration concentrates, how strongly it manifests, and when it breaks.
  • Auditable Alignment Signature: Aligned checkpoints show a terminal inflection in which αℓ rises while Lℓ falls, forming a dense spine where preference corrections accumulate.Unaligned baselines lack this signature.

3 The SPINAL Framework — Detecting Alignment via Geometric Fingerprints

SPINAL diagnoses alignment as a depth-indexed geometric trajectory rather than a single scalar, combining layerwise spectral scaling and semantic transition cost. Its operational claim is that alignment is a localized geometric calibration concentrated in terminal layers, where representations sharpen, belief transport decreases, paths stabilize, and optimization signal accumulates.

  • Core representation: SPINAL represents each checkpoint as a depth-indexed trajectory with coupled layerwise signals for spectral scaling and semantic transition cost.The trajectory is encoded by layer fingerprints gℓ = (ℓ, αℓ, Lℓ), capturing depth-wise semantic reorganization.
  • Geometric localization: Preference-induced geometric changes concentrate in final decoder blocks, where spectral changes dominate and the alignment effect is localized.The framework attributes this localization to preference gradients most strongly shaping the output distribution in the final decoder blocks.
  • Spectral scaling: The spectral exponent αℓ measures representational concentration: larger values concentrate mass at small singular-value indices, reduce effective dimension, and strengthen focus.SPINAL fits a power-law tail because layer activations often exhibit heavy-tailed spectra and preference sharpening is expected to steepen that tail.
  • Belief transport: The transport cost Lℓ measures adjacent-layer belief movement using Fisher–Rao geometry, with alignment expected to reduce terminal-block transport needed to reach the final predictive state.Beliefs are obtained through a logit lens and compared using a bounded Bhattacharyya-based Fisher–Rao step.
  • SPINAL diagnostic: Alignment is operationalized as terminal sharpening, reduced Fisher–Rao transport, stabilized trajectory coherence, and increased terminal optimization footprint.The combined diagnostic integrates contraction, coherence, and optimization-footprint signals; its ranking remains stable for at least 90% of random simplex weight draws.

4 Summary: SPINALScore Across Models

SPINALScore aggregates terminal-block sharpening, trajectory coherence, and optimization localization to quantify localized calibration across aligned models. Higher scores generally coincide with lower harmful compliance and higher refusal quality, while terminal-block disruption erases the geometric signature.

  • SPINALScore across aligned model families: SPINALScore combines terminal sharpening–contraction, trajectory coherence, and terminal gradient concentration; higher values indicate stronger terminal calibration.The diagnostic tracks ∆align, C(21:30)coh, and Gterm across the terminal decoder blocks.
  • SPINALScore across aligned model families: Phi-2 and Gemma show the clearest terminal calibration, Llama 3 and DeepSeek follow closely, and Qwen is milder but consistent.This ordering reflects calibration strength and localization rather than downstream safety or utility.
  • Behavior–geometry linkage: Higher SPINALSCORE is associated with lower HCR and higher SRQ, whereas HELP varies mainly with model family, scale, and instruction-tuning style.Across 10 Base and Aligned variants, Spearman ρHCR = −0.85, ρSRQ = +0.89, and ρHELP ≈0.05.
  • Interpretation and limitation: SPINALSCORE is computed purely from internal geometry and is not intended as a calibrated safety predictor.Its behavioral associations are presented as secondary linkage rather than a direct safety prediction.
  • Phi-2 ablations: removing/diffusing terminal alignment: Randomizing the terminal block, removing the preference objective, or diffusing updates erases the terminal fingerprint and collapses ∆align.Terminal Fisher–Rao cost P30ℓ=21 eLℓ increases, while terminal randomization produces high-curvature, irregular trajectories.

5 Conclusion

SPINAL makes alignment measurable across depth and shows that DPO’s geometric changes concentrate in a terminal calibration zone within the final decoder blocks. It captures this pattern through spectral sharpening, reduced Fisher–Rao belief transport, terminal coherence, and an aggregate auditing score.

  • SPINAL is a geometry-first diagnostic that makes model alignment measurable across depth.
  • DPO alignment concentrates in a terminal calibration zone within the final decoder blocks rather than diffusing across layers.
  • Aligned models show terminal spectral sharpening (↑αℓ), reduced Fisher–Rao belief transport (↓Lℓ), and terminal coherence.
  • SPINAL Score aggregates sharpening–contraction, trajectory coherence, and optimization concentration into one auditing score.

6 Discussion

SPINAL is a mechanistic diagnostic that represents checkpoints as depth-indexed curves to locate how preference optimization reshapes Transformer geometry near the output interface. Its terminal-block signature combines spectral sharpening, reduced belief transport, and increased optimization coherence to explain alignment localization.

  • Discussion: SPINAL treats each checkpoint as a depth-indexed curve in a two-dimensional state space, locating preference-induced geometric changes near the output interface.It is a diagnostic rather than a new alignment algorithm.
  • Discussion: The terminal-block signature jointly combines spectral sharpening, reduced belief transport, and increased coherence and optimization concentration in the last decoder layers.These signals are presented as a unified mechanistic account of alignment localization.
  • Discussion: Larger α_ℓ concentrates activation energy into fewer dominant directions, making representations more anisotropic and effectively lower-dimensional.The effective-dimension proxy decreases as α_ℓ increases.
  • Discussion: Smaller L_ℓ in terminal layers indicates fewer belief-state jolts as predictive distributions approach the output interface.Fisher–Rao distance measures movement on the probability simplex rather than Euclidean hidden-state change.
  • Discussion: Summing over ℓ∈[L −9, L] operationalizes the hypothesis that final layers are the calibration zone where preference gradients most directly determine output distributions.The resulting signed Δ_align increases when DPO produces terminal focusing together with terminal smoothing.

4. Why coherence and gradient concentration

SPINAL is presented as a diagnostic signature of terminal-layer calibration, but its scalar score supports screening rather than causal claims. Its localization may depend on architecture, alignment objective, theoretical assumptions, and evaluation distribution.

  • Diagnostic interpretation: SPINALSCORE is a localization index for focused, smooth, optimization-consistent calibration concentrated in final blocks.It aggregates SPINAL components into a scalar for reporting, but different terminal peaks, onset abruptness, or component trade-offs can produce similar totals.
  • Causality: SPINAL identifies a correlational terminal calibration signature associated with stronger alignment, not proof that terminal layers cause aligned behavior.Proposed causal tests include activation patching, causal tracing, and targeted layer ablations.
  • Objective dependence: Whether SPINAL transfers across DPO, RLHF, Constitutional, and reward-based objectives remains open because objectives can alter localization depth and dominant components.The outcome determines whether a universal SPINALSCORE or objective-aware summaries are appropriate.
  • Theoretical grounding and sensitivity: The measured geometry is rigorous, but its thermodynamic interpretation remains provisional because layerwise logit changes do not automatically define a thermodynamic process.Interpretive risks include ambiguous contraction mechanisms, unproven score invariances, prompt-distribution shifts, and token-position dependence.

7 Frequently Asked Questions (FAQs) … A.6 Terminal alignment delta ∆align and SPINALSCORE aggregation

SPINAL is a protocolized localization diagnostic, not a causal theorem, behavioral predictor, or safety certificate. It combines spectral sharpening, Fisher–Rao contraction and coherence, and terminal optimization localization into decomposed per-layer diagnostics plus a comparative SPINALScore.

  • 7 Frequently Asked Questions (FAQs): SPINAL does not establish causality, replace behavioral suites, predict any single metric, or certify deployment safety.Causal validation would require targeted terminal interventions, while behavioral suites remain necessary for external validation across task distributions and deployment regimes.
  • 7 Frequently Asked Questions (FAQs): SPINAL identifies a repeatable terminal-block signature that co-varies with alignment-tuned checkpoints under a fixed measurement protocol.The signature includes spectral tail sharpening, contraction of successive layerwise distributions, and localization of optimization signal in a terminal window.
  • 7 Frequently Asked Questions (FAQs): SPINAL is intended primarily for screening and debugging paired base–aligned checkpoints, before expensive behavioral evaluation.Per-layer trajectories can reveal loss of contraction or dispersion of the terminal footprint after merging, quantization, distillation, or continued tuning.
  • 7 Frequently Asked Questions (FAQs): The Fisher–Rao step length provides a canonical information-geometric scale for comparing adjacent layerwise categorical distributions under a fixed protocol.It is computed from the Bhattacharyya coefficient as L(p, q) = 2 arccos(BC(p, q)).
  • 7 Frequently Asked Questions (FAQs): SPINAL is objective-agnostic: different alignment objectives may produce different localization patterns, which the diagnostic is designed to reveal.For example, RLHF could shift contraction or coherence earlier or make them multimodal across depth rather than terminally localized.
  • A.5 Terminal optimization footprint Gterm (alignment localization): Large Gterm indicates that a substantial fraction of the optimization signal is absorbed by the terminal block, consistent with depth-localized alignment updates.The terminal footprint is one of three distinct views of terminal calibration, alongside spectral and distributional diagnostics.
  • A.6 Terminal alignment delta ∆align and SPINALSCORE aggregation: SPINALScore aggregates terminal sharpening–contraction, terminal coherence, and terminal optimization footprint because their conjunction probes non-redundant objects.The scalar supports checkpoint comparison and triage, while decomposed components and per-layer curves preserve mechanistic interpretability.

A.7 Reproducibility … B.4 Practical computation: truncation, renormalization, and geometric meaning.

SPINAL requires a locked measurement pipeline and formalizes layerwise belief transport with Fisher–Rao geometry, using Hellinger embeddings and Bhattacharyya overlap. Its practical implementation uses fixed top-k truncation and renormalization while documenting numerical safeguards and interpretive limits.

  • A.7 Reproducibility: SPINAL fixes prompts, batching, token positions, randomness, spectral fitting, and Fisher–Rao approximation choices to make cross-checkpoint measurements reproducible.The default uses the prefill-last-token position with dropout disabled and fixed seeds.
  • A.7 Reproducibility: Reproducibility artifacts include prompts, seeds, layer-index conventions, logit-lens specifications, and gradient-share logs for reproducing per-layer curves and aggregate scores.These artifacts are described as small compared with model weights.
  • B Information geometry of belief transport (Fisher–Rao + Bhattacharyya): Each layer induces a categorical next-token distribution, and SPINAL measures adjacent-layer movement on the probability simplex using Fisher–Rao geometry.The resulting step length is canonical, computationally stable through Bhattacharyya overlap, and comparable across checkpoints under a fixed metric.
  • B.1 Probability simplex and the Fisher infor-: Fisher–Rao distance is invariant under smooth reparameterizations, making it independent of arbitrary coordinates for layerwise distributions on the simplex.This intrinsic property motivates its use for comparing categorical beliefs induced at different layers.
  • B.2 The Hellinger embedding and the spherical (angle) form.: The square-root Hellinger embedding maps distributions to the unit sphere, where Fisher–Rao geodesic distance becomes the spherical angle 2 arccos(BC(p, q)).The Bhattacharyya coefficient lies in [0, 1], so the distance is bounded in [0, π].
  • B.3 From distance to transport: layerwise step length and path length.: SPINAL aggregates adjacent-layer Fisher–Rao steps across prompts and token positions, with normalized transport eLℓ = Lℓ/π ∈ [0, 1].Aligned checkpoints exhibit terminally contractive behavior, in which late layers move induced distributions less; this is diagnostic rather than causal.
  • B.4 Practical computation: truncation, renormalization, and geometric meaning.: The default top-k approximation retains kFR = 2048 tokens, then renormalizes their probabilities before computing the truncated Bhattacharyya coefficient.Renormalization restores unit norm in the Hellinger embedding and preserves the spherical-geodesic interpretation.
  • B.4 Practical computation: truncation, renormalization, and geometric meaning.: Increasing kFR increases captured mass and decreases Fisher–Rao error, while clamped and unclamped implementations nearly overlap across a wide range of kFR.The protocol recommends choosing the smallest kFR that meets the required accuracy and reporting truncation behavior.

B.7 Implementation-level guidance (protocol … C.0.2 Diagnostics: what to plot and what to check

SPINAL requires fixed, numerically stable measurement protocols and treats the spectral tail exponent αℓ as a local comparative diagnostic rather than evidence of a universal power law. Valid interpretation depends on strict fitting criteria, stability checks, and diagnostic plots that test residual structure, window sensitivity, null baselines, and cross-layer coherence.

  • B.7 Implementation-level guidance (protocol: SPINAL recommends deterministic prefill measurements, fixed top-k truncation with captured mass reporting, fp32 Bhattacharyya computation, clamping before arccos, and π-normalized distances.These commitments improve reproducibility, numerical stability, approximation transparency, and interpretability.
  • B.7 Implementation-level guidance (protocol: The Fisher–Rao construction is the canonical statistical-manifold metric, with Bhattacharyya and Hellinger-angle closed forms enabling reproducible belief-transport measurements.SPINAL uses this geometry to quantify terminal contraction as a grounded signature of aligned checkpoints.
  • C Spectral tail exponent αℓ: fitting protocol and diagnostics: αℓ is a protocol-defined, windowed spectral-shape descriptor, not evidence that the entire spectrum follows a universal power law.Layers failing fit criteria remain undefined rather than being forced into the analysis.
  • C Spectral tail exponent αℓ: fitting protocol and diagnostics: Mean-center activations, compute their eigenspectrum or singular-value spectrum, and estimate αℓ from a log–log slope over a designated tail window.Mean-centering prevents a trivial global-shift component from contaminating the apparent tail, while eigenvalue and singular-value fits are equivalent up to log-space constants.
  • C.0.1 Complete tail-fit procedure (protocol: The complete procedure fixes prompts, token selection, sample size, preprocessing, and reported (N, d), then applies positive-eigenvalue safeguards and anti-cherry-picking window rules.Candidate windows use a minimum length, while fixed tail lengths or fixed fractional ranges are recommended across experiments.
  • C.0.1 Complete tail-fit procedure (protocol: A window is accepted only when R2 ≥τR2, |K|≥mmin, and ˆsℓ< 0; selection then maximizes R2 or minimizes residual MSE while avoiding the numerical floor.Recommended stability estimation repeats the procedure across S subsamples, and large stability error makes the layer estimate unreliable for claims.
  • C.0.2 Diagnostics: what to plot and what to check: Diagnostics should plot the fitted log–log segment, inspect residual curvature, test window-sensitivity plateaus, compare matched random-matrix baselines, and assess coherent cross-layer trends.Rapid window changes, unstable null matches, or isolated spikes can indicate underdetermination, non-model-specific geometry, rank collapse, insufficient N, or numerical-floor fitting.

C.0.3 Failure modes and exclusion criteria (to prevent over-interpretation) … D.5 Recommended reporting template (for comparability and auditability)

SPINAL is designed as a conservative, componentized diagnostic: invalid spectral fits or unstable Fisher–Rao estimates are excluded, while interpretable geometry, coherence, and optimization signals are reported for auditability. Comparability requires explicit protocols, baselines, depth curves, terminal scalars, and diagnostic exclusions.

  • C.0.3 Failure modes and exclusion criteria (to prevent over-interpretation): Insufficient rank, numerical-floor dominance, multi-regime spectra, and mean-shift or outlier effects can invalidate tail-shape estimation and require exclusion or diagnostic mitigation.The protocol rejects insufficient effective rank, limits kmax relative to numerical floors, checks fit residuals and R2, and recommends centering and stability checks.
  • C.0.3 Failure modes and exclusion criteria (to prevent over-interpretation): Window search can create selection bias, so the protocol fixes m or fractional bounds and reports the selection policy and number of windows searched.This addresses the risk of finding an apparently linear segment by accident when too many candidate windows are tested.
  • C.0.4 Reproducibility checklist (reporting template): αℓ is a protocol-defined spectrum-shape descriptor, not evidence of a generative power-law mechanism, and failed fits are treated as undefined rather than imputed.Within SPINAL, αℓ is used comparatively across layers and checkpoints to track relative spectral sharpening or flattening.
  • D.5 Recommended reporting template (for comparability and auditability): For comparability, report full-depth curves, terminal-block scalars, explicit protocol settings, matching randomized or Gaussian baselines, and conservative exclusions for unreliable αℓ or Lℓ.The reporting template includes Fisher–Rao stability artifacts and requires explicit sampling, preprocessing, terminal-block, and gradient-estimation policies.
  • D SPINAL components and SPINALScore construction: SPINAL decomposes alignment evidence into terminal sharpening–contraction, coherence, and terminal optimization footprint before forming an auditable aggregate score.The componentized design lets reviewers inspect which mechanism drives a high score and whether it is numerically stable and behaviorally meaningful.
  • D.2 Component (ii): terminal coherence: Terminal coherence is high when Fisher–Rao step lengths are smooth and stable, because normalized total variation penalizes jagged or oscillatory terminal trajectories.The coherence score is defined as coh = exp(−γ TVterm), γ > 0.
  • D.2 Component (ii): terminal coherence: Gterm measures the fraction of total gradient energy in the terminal block, with Gterm ≈1 indicating primarily terminal optimization and small values indicating distributed learning.Gradient magnitudes should be estimated across minibatches, with parameter-count normalization available to avoid bias toward larger layers.
  • D.4 Component (iv): aggregating into SPINALScore: SPINALScore combines normalized component evidence so geometry coupling, stabilization, and optimization localization jointly support a high score, while low scores remain diagnostically interpretable.Low scores can reflect weak sharpening–contraction coupling, oscillatory terminal geometry, or distributed optimization.

E Reproducibility protocol and artifact commitments … Minimum replication recipe (what an independent team should do)

The reproducibility protocol fixes measurement choices and commits to releasing the prompts, model identities, code, environments, results, and reporting metadata needed for faithful replication. Independent teams should regenerate per-layer curves and SPINALScore, inspect stability diagnostics, and compare outputs within stated tolerances.

  • E Reproducibility protocol and artifact commitments: SPINAL treats reproducibility as a protocol because silent choices about prompts, tokens, truncation, clamping, terminal windows, and randomness can change measurement outcomes.The pipeline combines layerwise spectra, Fisher–Rao step-lengths, and terminal aggregation.
  • Protocol Box →concrete checklist of fixed defaults: Reported SPINAL curves and SPINALScore use fixed defaults so independent teams obtain the same curves up to floating-point tolerance.Defaults may be overridden only in a released configuration.
  • Protocol Box →concrete checklist of fixed defaults: The protocol fixes a prompt pool, keeps |X| constant across models, explicitly reports the token rule, and defaults to evaluating only the final position after KV-cache prefill.The last-token prefill reduces ambiguity about token subsampling and runtime variance.
  • Protocol Box →concrete checklist of fixed defaults: It also fixes the terminal window and makes Fisher–Rao truncation a first-class parameter, with probability clamping, renormalization, and reported stability over ε and τ.The default ε is a small constant such as 10^-12, while τ is close to 1, such as 0.999.
  • Artifact commitments: what we must release for faithful replication: Artifact commitments cover prompt text, stable IDs and provenance, exact tokenization, checkpoint revisions and hashes, precision, inference backends, end-to-end code, tests, plotting, and environments.Released environment metadata includes hardware, software versions, determinism settings, and a container recipe.
  • Artifact commitments: what we must release for faithful replication: Result artifacts include raw per-layer arrays, SPINALScore and component scalars, and scripts, configurations, and checksums for exact figure regeneration.These releases provide direct comparison targets for replication.
  • Minimum replication recipe (what an independent team should do): A faithful replication loads the released prompts and model revision, runs the released configuration, produces curves, SPINALScore, kFR and tail-fit diagnostics, and compares outputs within stated tolerance.The diagnostics include clamp/mass sensitivity and an R2 pass/fail illustration.
  • Minimum replication recipe (what an independent team should do): Every reported SPINALScore must disclose prompt-pool identity and |X|, token rule, seeds, stability repeats S, model hash, and environment versions; unreproducible metrics are nonscientific.The commitment applies to main-text SPINALScore values and supports artifact evaluation.

F Experimental setup: checkpoints, prompts, compute, and evaluation suites

The experimental contract pairs immutable Base and Aligned checkpoints, fixes prompts and runtime conditions, and requires artifact disclosure sufficient to reproduce SPINAL curves, scores, and optional behavioral probes. Evaluation separates deterministic geometry measurement from generation-based behavior, with replication judged against stated tolerances under identical artifacts and runtime.

  • F.1 Checkpoints and pairing protocol: Each model family and size is evaluated as a paired Base pretrained checkpoint versus an Aligned instruction-tuned checkpoint, with preference optimization treated as a recipe label.Pairs should match architecture, tokenizer, context window, and inference stack; any violation is flagged because token-boundary changes can alter activation statistics.
  • F.1 Checkpoints and pairing protocol: Every checkpoint records a Hub ID, commit SHA or release tag, weight hash, tokenizer hash, and loader/inference code hash; any difference defines a different experiment.The required roster lists family, size, Base versus Aligned status, alignment objective label, and source identifiers.
  • F.2 Inference/runtime regime (precision, batching, determinism): SPINAL measurement and behavioral probes use separate regimes: deterministic prefill-only activation measurement versus explicitly configured greedy or stochastic generation and scoring.Measurement runs disable decoding and dropout, use eval mode and recorded seeds, while precision, kernels, accumulation, layernorm, stability, batching, and token positions are fixed and reported.
  • F.3 Compute and hardware (what must be reported): Hardware disclosure covers GPU type and count, memory and interconnect, CPU, RAM, operating system, and driver/CUDA versions because numerical choices affect spectral tails and Fisher–Rao computations.Recommended compute reporting includes wall-clock time, effective throughput, and peak GPU memory.
  • F.4 Prompt pool(s): composition, IDs, and release format: Prompt pools are versioned first-class datasets with stable IDs, normalized exact text, metadata, stratification, published sampling rules, and mandatory JSONL and split-seed releases.Stratification includes domain, safety versus benign status, and prompt-length bins; prompts.jsonl contains id, text, domain, safety_flag, len_bin, source, and notes.
  • F.5 Evaluation suites and behavioral probes (if reported): Reported outcomes comprise geometry-only curves and SPINALScore, or geometry plus behavioral probes whose prompt sets, generation settings, scoring rules, and evaluator identity are disclosed.Model-based evaluators follow the same immutability contract as other experimental models.
  • F.6 What must be released (artifact commitments): Replication requires releasing checkpoint and prompt identifiers, computation scripts, system logs, and run manifests, with per-layer curves and SPINALScore matching stated tolerances under identical artifacts and runtime.Example tolerances are mean absolute deviation ≤δ on the terminal window and ±0.02 in normalized SPINALScore units.

G Robustness and sensitivity analyses (measurement stability) … G.5 (iv) Terminal window selection sensitivity

SPINAL’s robustness protocol tests whether conclusions persist under perturbations to prompts, token positions, Fisher–Rao truncation, and terminal windows. It evaluates both absolute score stability and comparative rank stability, with failure requiring explicit diagnosis or restriction rather than tuning.

  • G Robustness and sensitivity analyses (measurement stability): SPINAL treats robustness as a protocol commitment, requiring preserved checkpoint rankings and small absolute-score changes under measurement perturbations.The goal is to make SPINAL a measurement rather than an artifact of hyperparameters.
  • G.1 What we mean by stability: Stability is assessed through relative score deviation and Spearman rank correlation across controlled protocol instances.The protocol samples perturbations over prompt subsampling, token rules, Fisher–Rao settings, and terminal windows.
  • G.2 (i) Prompt distribution and subsampling sensitivity: Prompt subsampling evaluates score means, standard errors, and rank stability across replicated subsets drawn from a fixed master prompt pool.Recommended acceptance targets are ρrank ≥ 0.9 and RelDev(m) < 5% for most checkpoints, with outliers reported.
  • G.3 (ii) Token position choice sensitivity: Token-position sensitivity compares default last-token prefill measurements with averages over a short greedy continuation.The comparison uses ∆tok(m) and Spearman ρrank, while diagnosing shifts in geometry, spectral, or optimization components.
  • G.4 (iii) Fisher–Rao truncation sensitivity: kFR and captured mass: Fisher–Rao sensitivity varies captured mass thresholds τ ∈ {0.95, 0.975, 0.99, 0.995} and clamp floors ε ∈ {10^-12, 10^-10, 10^-8}.Downstream effects are measured through SPINALScore(m) and rank-order ρrank.
  • G.4 (iii) Fisher–Rao truncation sensitivity: kFR and captured mass: Fisher–Rao instability is flagged by abrupt kFR jumps, heavy-tailed step-length outliers, or disagreement between Fisher–Rao and overall score stability.The recommended response is to tighten the default regime, such as using higher τ, and report the restriction.
  • G.5 (iv) Terminal window selection sensitivity: Strong dependence on terminal-window width, including rank-order flips, indicates that late localization is unsupported and requires broadening or restricting the claim.The terminal block is treated as a hypothesis because SPINAL assumes alignment-induced calibration is localized late.
  • G.5 (iv) Terminal window selection sensitivity: Terminal-window sensitivity tests W(w) over w ∈ {6, 8, 10, 12}, optionally shifting the window by δ ∈ {0, 1, 2}.It compares SPINALScore(m; W(w)), component attribution, and Spearman ρrank across windows.

G.6 Concise robustness checklist (protocol-grade) … H.5.3 CV3: Causal tracing / targeted edits (factored mechanisms)

The appendix specifies a conservative, reproducible robustness protocol for SPINAL, extends checkpoint reporting and controls against geometry mirages, and adds qualitative and optional causal-validation analyses. Together, these procedures constrain interpretation by pairing geometry with behavior and testing whether terminal signals are stable, specific, and causally relevant.

  • G.6 Concise robustness checklist (protocol-grade): SPINAL releases must fix prompt pools, token rules, terminal windows, seeds, and determinism flags across runs, then report sensitivity statistics and explicit pass criteria.The recommended protocol emphasizes reproducibility and conservative claim narrowing when robustness fails.
  • H.1 Extended checkpoint sweep: breadth, pairing, and reporting: Extended evaluations should use paired checkpoints within family and size, reporting per-layer curves, SPINALScore components, fit-validity flags, and evaluator-sensitive probe regimes.Pairing is intended to isolate alignment effects from architecture or scale changes, while flags expose truncation, fit, and exclusion issues.
  • H.2 Controls and ablations: ruling out “geometry mirages”: Controls are methodological requirements because prompt drift, numerical truncation, tail-window selection, and terminal heuristics can create appealing but misleading geometry.The appendix treats these controls as part of SPINAL rather than as post hoc validation.
  • H.2.1–H.2.5 Controls and ablations: Robust signatures should preserve pair rankings and component direction across terminal windows and prompt pools, while token, kFR, and tail-fit changes must not create unexplained inversions.Window shifts that flip signs can indicate boundary artifacts, insufficient samples, or numerical floors; prompt-sensitive deltas should be disclosed as limitations.
  • H.3 Specificity checks: does SPINAL measure “alignment” or “anything”?: Specificity checks compare alignment with domain, style, helpfulness, precision, and terminal-noise controls; quantization may require larger kFR, but SPINALScore should not spuriously mimic alignment.Unstructured terminal noise is expected to degrade coherence and inflate Fisher–Rao step length without reproducing alignment’s coupled signature.
  • H.4 Qualitative analysis: success modes, failure modes, and edge cases: Qualitative success cases combine correct helpful refusals, stable benign helpfulness, reduced harmful compliance, and terminal localization, whereas safety-blanket and policy-hole cases expose opposite behavioral and geometric failure modes.Over-application can increase coherence while reducing helpfulness; under-application is associated with weak localization and elevated terminal Fisher–Rao steps.
  • H.5 Optional causal-validation protocol (forward-looking, testable): The optional causal-validation blueprint uses activation patching, path patching, and causal tracing to test whether terminal features and specific attention, MLP, residual, or internal states mediate behavior.Reliable transfer from terminal-layer patching would support terminal localization; dependence on earlier layers would make that story incomplete.

H.5.4 CV4: Reporting standards for causal validation

Causal validation should report enough procedural detail and artifacts to make interventions reproducible. These protocols are optional but connect geometry diagnostics to mechanism-level evidence when undertaken.

  • Reproducibility requirements: Report exact prompt variants, corruption methods, decoding settings, patched modules, and layer ranges.These details define the intervention and its scope.
  • Reproducibility requirements: Release patching scripts and model identifiers, as in Appendix E.The released artifacts allow others to reproduce the causal intervention.
  • Purpose: Causal protocols turn interpretability from narrative into interventional science when time permits.They create a bridge between geometry diagnostics and mechanism-level evidence, although they are optional.
  • Failure-mode documentation: Release prompt IDs and text, raw generations, seeds, model identifiers or hashes, and per-layer curves for each failure-mode case.Each case is specified as a reproducible unit test linking prompts, behavior, SPINAL signals, diagnosis, and control.

I Spinal Metrics … I.8 Sinkhorn Divergence: Transport-Length Proxy

SPINAL combines complementary representation and transport diagnostics to test whether preference alignment produces depth-localized geometric changes. Across these metrics, the expected signature is stable early-layer structure followed by terminal-layer focusing, deformation, stabilization, coherence, and reduced transport difficulty.

  • I.1 Effective Rank: Effective rank measures entropy-based spectral dimensionality, with lower values indicating variance concentrated into fewer semantic directions.It complements the spectral tail exponent by capturing the global distribution of spectral mass.
  • I.1 Effective Rank: Terminal effective-rank drops provide independent evidence of representation sharpening, while early-layer values are expected to remain comparatively stable.An effective rank near 1 indicates near-degeneracy, whereas values near r indicate broadly spread variance.
  • I.2 Centered Kernel Alignment (CKA): CKA compares checkpoint representations invariantly to isotropic scaling and orthogonal rotations, enabling localization of terminal reshaping.Low angular CKA distance indicates strong similarity, while terminal spikes indicate structural geometric change near the output interface.
  • I.3 Procrustes Distance: Procrustes distance removes optimal rotational alignment so residual terminal mismatch identifies alignment-induced deformation that changes representation shape.Low distance indicates rotation-equivalent differences; high terminal distance indicates non-rotation-equivalent deformation.
  • I.4 CKA Cross-Model Divergence: Layerwise CKA cross-model divergence should remain near zero early and increase sharply in the terminal window under localized alignment.This curve directly tests the transition from a shared backbone to an aligned interface.
  • I.5 L2 Norm Change: Late-layer decreases in average per-token L2 displacement indicate smaller refinements rather than large representational jolts during aligned stabilization.The metric acts as a layer-to-layer activation-space step-size diagnostic.
  • I.6 Activation Norm: Activation norms serve as a sanity check: comparable norms support interpreting CKA, Procrustes, and spectral changes as structural rather than rescaling artifacts.Large deviations flag potential numerical confounds.
  • I.7 Projection Norm: Higher terminal projection norm indicates a shared directional correction across samples, supporting structured rather than noisy geometric transformation.Terminal directional coherence is expected when aligned models undergo localized calibration.
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